Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518904 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2408 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,14)(11,12)(17,18)(19,25)(20,28)(21,24)(22,26)(23,29)(27,30)(31,32);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(19,26)(20,27)(22,25)(23,31)(28,30)(29,32);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,19,21,26)(18,22,24,25)(20,27,29,32)(23,30,28,31);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,15,14,16)(17,20,19,27,21,29,26,32)(18,23,25,31,24,28,22,30);(2,8)(6,7)(9,12,13,11)(10,15,14,16)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=fghi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72340544, #72340548, #72340627, … |
| maximal normal subgroup of | #72340544, #72340548, #72340627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7527879);